The only thing I can find is that the power supply takes 5 - 12v dc or up to 20v at the max.

The processor and other circuitry on your Mega all runs at 5v. There is an on-board regulator that takes your 12v supply and regulates it down to 5v for the processor.

The problem is that the type of regulator used on the Mega is not very efficient.. To get from 12v to 5v, it has to burn off more than half the power as heat. That is not a big deal if you are running the Mega by itself, since the Mega doesn't use a lot of power. But when you start adding a lot of other circuitry, the amount of power that has to be dumped as heat increases as well. At some point, the regulator will overheat and shut itself down.

LED strips are pretty power hungry and they run off 5v too. I would recommend giving them their own supply to ease the burden on your Mega's regulator. This would be a good choice: http://www.adafruit.com/products/276

That depends on how hard you are driving your LED strips. With all LEDs on full brightness white, the strip will need close to the full 2A, so 2 would be advisable. We also have a 5v 10A supply which would be enough to power both strips, plus all your other circuitry. http://www.adafruit.com/products/658 Unfortunately, they are out of stock at the moment.

would I be okay since the LEDs run off the 5v rail on the Mega?

Don't connect your leds to the 5v rail on the Mega. To wire these up, connect all of the grounds together (Arduino and both LED strips). But keep the 5v lines separate.A couple of these adapters will make your wiring a bit easier: http://www.adafruit.com/products/368

Can I run the Nintendo controllers off the Mega or should I add those to the LED power supply as well? I have no idea the current draw for them but I can look it up. As long as the LEDs and the controllers are less than 2 amps I should be ok yes?

The power to the LEDs goes to the power supply and then a ground pin on the Mega correct? I think I understand now.

Just googling some more to make sure I have this whole thing correct and I found this

5V. This pin outputs a regulated 5V from the regulator on the board. The board can be supplied with power either from the DC power jack (7 - 12V), the USB connector (5V), or the VIN pin of the board (7-12V). Supplying voltage via the 5V or 3.3V pins bypasses the regulator, and can damage your board. We don't advise it.